Theorem 2.5 ([1]}, Theorem 2.10) 
Let \(X\)  be a smooth projective surface over a field \(k\) .
Let \(D\)  be a \(\mathbb {Z}\)  divisor on \(X\) .
Then \(\chi (X,D)=\chi (X,O_X)+\frac{1}{2}D\cdot _k(D-K_X)\) .

